Abstract

I implement the genetic algorithm and simulated annealing to optimise the operation of a pumped storage system in the day-ahead market. How do they compare with the solution found by classical mixed-integer linear programming? Let’s find out.

# Data processing
import numpy as np
import pandas as pd
import polars as pl

# Optimisation
from pulp import (
    LpMaximize,
    LpMinimize,
    LpProblem,
    LpStatus,
    lpSum,
    LpVariable,
    LpContinuous,
    LpInteger,
    value
)


# Visualisation
import plotnine as pn
import plotly.express as px
from IPython.display import display, clear_output
from mizani.formatters import comma_format, percent_format
import imageio

# Hyperparameter tuning
import optuna
import ray
from ray import train, tune
from ray.tune.search.optuna import OptunaSearch
from ray.tune.schedulers import ASHAScheduler

# Utils
import os
from tqdm.notebook import tqdm
import json
from itertools import product
import warnings

# Options
background_colour = "#F2F2F2"
pn.theme_set(
    pn.theme_classic()
    + pn.theme(
        text=pn.element_text(family="monospace"),
    )
)
pd.set_option('display.float_format', lambda x: '%.3f' % x)

%load_ext blackcellmagic

TLDR

If you’re short on time or disinterested, here’s the essence: The advantages of heuristic (meaning: including trial and error) optimisation methods is that they can deal with highly complex optimisation problem involving non-linear objective functions and still find a decent solution where classical optimisation might fail to find the optimal one. The goal of this blog post is to benchmark these methods on a problem which is quite easy to solve using linear programming, i.e. one of the classical approaches. What will be the price to pay when using heuristics instead?

As shown in the animation above, neither the genetic algorithm, nor simulated annealing were able to find the optimal solution (350k of profit), but they were very close. The computing time allowed for both (also for random search) was 30 seconds. Given the simplicity of these algorithms, I really find this result amazing and I am positively surprised by how well they did.

Problem setting: The pumped storage hydropower plant

TBD: Sketch decisions in Excalidraw

We have a week’s worth of data, so we have 168 decisions to make on the hourly time scale.

The revenue at each hour is:

\[dt_i * P_{Turbine} * Spot_i + dp_i * P_{Pump} * Spot_i\]

plant_params = {
    "EFFICIENCY": 0.75,
    "MAX_STORAGE_M3": 5000,
    "MIN_STORAGE_M3": 0,
    "TURBINE_POWER_MW": 100,
    "PUMP_POWER_MW": 100,
    "TURBINE_RATE_M3H": 500,
    "MIN_STORAGE_MWH": 0,
    "INITIAL_WATER_LEVEL_PCT": 0,
}
plant_params["INITIAL_WATER_LEVEL"] = (
    plant_params["INITIAL_WATER_LEVEL_PCT"] * plant_params["MAX_STORAGE_M3"]
)
plant_params["PUMP_RATE_M3H"] = (
    plant_params["TURBINE_RATE_M3H"] * plant_params["EFFICIENCY"]
)
plant_params["MAX_STORAGE_MWH"] = (
    plant_params["MAX_STORAGE_M3"] / plant_params["TURBINE_RATE_M3H"]
) * plant_params["TURBINE_POWER_MW"]

plant_params

{'EFFICIENCY': 0.75,
 'MAX_STORAGE_M3': 5000,
 'MIN_STORAGE_M3': 0,
 'TURBINE_POWER_MW': 100,
 'PUMP_POWER_MW': 100,
 'TURBINE_RATE_M3H': 500,
 'MIN_STORAGE_MWH': 0,
 'INITIAL_WATER_LEVEL_PCT': 0,
 'INITIAL_WATER_LEVEL': 0,
 'PUMP_RATE_M3H': 375.0,
 'MAX_STORAGE_MWH': 1000.0}

Data

TBD describe

prices = (
    pl.read_csv("01 - Data/spot_prices_utc.csv")
    .with_columns(pl.col("utc_time").str.to_datetime(), pl.col("spot").forward_fill())
    .drop_nulls()
)

prices.glimpse()

Rows: 48166
Columns: 2
$ spot                   <f64> 36.99, 31.08, 29.17, 21.96, 14.96, -5.92, -6.1, -10.02, -5.71, -1.88
$ utc_time <datetime[μs, UTC]> 2017-12-31 23:00:00+00:00, 2018-01-01 00:00:00+00:00, 2018-01-01 01:00:00+00:00, 2018-01-01 02:00:00+00:00, 2018-01-01 03:00:00+00:00, 2018-01-01 04:00:00+00:00, 2018-01-01 05:00:00+00:00, 2018-01-01 06:00:00+00:00, 2018-01-01 07:00:00+00:00, 2018-01-01 08:00:00+00:00

window_size = 24*7

example = prices.filter(
    pl.col("utc_time") >= pd.to_datetime("2022-01-01").tz_localize("UTC")
).head(window_size)

example.head()

shape: (5, 2)

spot	utc_time
f64	datetime[μs, UTC]
101.54	2022-01-01 00:00:00 UTC
52.13	2022-01-01 01:00:00 UTC
20.78	2022-01-01 02:00:00 UTC
15.66	2022-01-01 03:00:00 UTC
21.47	2022-01-01 04:00:00 UTC

example.shape

(168, 2)

Genetic Algorithm

Fitness Function

TBD

def evaluate_fitness(population, ps_params, prices):

    # To be written according to pumped storage optimisation problem
    fitness_scores = np.array([])

    for parameter_combination in population:

        water_level = ps_params["INITIAL_WATER_LEVEL"]
        fitness_score = 0

        for action, price in zip(parameter_combination, prices["spot"]):
            # Pump (-1)
            if action == -1:
                if (
                    water_level + ps_params["PUMP_RATE_M3H"]
                    <= ps_params["MAX_STORAGE_M3"]
                ):
                    fitness_score -= ps_params["PUMP_POWER_MW"] * price
                    water_level += ps_params["PUMP_RATE_M3H"]
                else:
                    fitness_score -= 100_000
            # Turbine (1)
            if action == 1:
                if (
                    water_level - ps_params["TURBINE_RATE_M3H"]
                    >= ps_params["MIN_STORAGE_M3"]
                ):
                    fitness_score += ps_params["TURBINE_POWER_MW"] * price
                    water_level -= ps_params["TURBINE_RATE_M3H"]
                else:
                    fitness_score -= 100_000
            # Do nothing (0)
            # Nothing happens to the fitness score and the water level

        fitness_scores = np.append(fitness_scores, fitness_score)

    return fitness_scores

The Algorithm

class GA_discrete_actions:

    def __init__(
        self,
        dna_size,
        discrete_action_space,
        elitism=0.25,
        survival_rate=0.25,
        population_size=200,
        initial_mutation_rate=0.1,
        final_mutation_rate=0.01,
        total_generations=100
    ):

        self.dna_size = dna_size
        self.discrete_action_space = discrete_action_space
        self.elitism = elitism
        self.survival_rate = survival_rate
        self.population_size = population_size
        self.elite_size = int(self.elitism * self.population_size)
        self.initial_mutation_rate = initial_mutation_rate
        self.mutation_rate = initial_mutation_rate
        self.final_mutation_rate = final_mutation_rate
        self.total_generations = total_generations
        
        self.best_dna = None
        self.best_fitness = None
        self.average_fitness = None

        self.initial_population = self.__create_random_population()

    def evolve(self, population, fitnesses):

        assert len(population) == self.population_size
        assert len(fitnesses) == len(population)
        assert isinstance(population, np.ndarray)

        # Sort population and fitnesses
        fitness_indices = fitnesses.argsort()
        sorted_fitnesses = fitnesses[fitness_indices]
        sorted_population = population[fitness_indices]

        # Report the best fitness and parameters
        self.best_dna = sorted_population[-1]
        self.best_fitness = sorted_fitnesses[-1]
        self.average_fitness = np.mean(sorted_fitnesses)

        # Only keep the elite of the population and get their weighting
        elite = sorted_population[-self.elite_size :]
        sorted_elite_fitnesses = sorted_fitnesses[-self.elite_size :]

        # Calculate weighting for elite
        min_fitness = sorted_elite_fitnesses.min()
        max_fitness = sorted_elite_fitnesses.max()

        if min_fitness == max_fitness:
            # If all fitnesses are equal, set weighting to uniform distribution
            fitnesses_weighting = np.full_like(
                sorted_elite_fitnesses, 1 / len(sorted_elite_fitnesses), dtype=np.float32
            )
        else:
            fitnesses_weighting = (sorted_elite_fitnesses - min_fitness) / (
                max_fitness - min_fitness
            )
            fitnesses_weighting /= fitnesses_weighting.sum()

        new_population = np.zeros((self.population_size, self.dna_size))

        # Keep the population according to survival rate
        survival_cutoff = int(np.floor(self.survival_rate * self.population_size))
        new_population[0:survival_cutoff] = sorted_population[-survival_cutoff:]

        # Crossover the elite only, rest of population doesn't cross over
        for child_id in np.arange(survival_cutoff + 1, self.population_size):
            i0 = np.random.choice(
                a=self.elite_size, p=fitnesses_weighting, replace=True
            )
            i1 = np.random.choice(
                a=self.elite_size, p=fitnesses_weighting, replace=True
            )

            new_dna = self.__crossover(elite[i0], elite[i1])
            
            assert isinstance(new_dna, np.ndarray)

            new_population[child_id] = new_dna

        # Mutate the new population
        for dna_id in range(self.population_size):
            new_population[dna_id] = self.__mutate(new_population[dna_id])

        # Adjust mutation rate: linear decay dependent on total generations
        self.mutation_rate = self.mutation_rate - 1/self.total_generations * (self.initial_mutation_rate - self.final_mutation_rate)
        self.mutation_rate = np.maximum(self.mutation_rate, self.final_mutation_rate)

        assert new_population.shape == self.initial_population.shape

        return new_population

    def __create_random_population(self):
        population = np.random.choice(
            a=self.discrete_action_space, size=(self.population_size, self.dna_size)
        )
        return population

    def __mutate(self, dna):

        # If random dice roll (between zero and one) is less than mutation
        # rate (between zero and one) then inject noise into the dna.
        if np.random.random_sample() < self.mutation_rate:
            # Randomly select which element to mutate and insert random integer parameters
            position = np.random.choice(a=np.arange(0, len(dna)), size=1)
            new_options = [x for x in self.discrete_action_space if x != dna[position]]
            mutation_value = np.random.choice(a=new_options, size=1)
            dna[position] = mutation_value

        return dna

    def __crossover(self, dna1, dna2):
        assert len(dna1) == len(dna2)
        # set child's DNA to be of the first parent
        child = np.copy(dna1)
        # replace random positions with dna from second parent
        dna2_indices = np.random.randint(2, size=child.size)
        indices = np.where(dna2_indices)
        child[indices] = dna2[indices]

        return child

ga = GA_discrete_actions(
    dna_size=window_size,
    discrete_action_space=[-1, 0, 1],
    elitism=0.5,
    population_size=500,
    survival_rate=0.25,
    initial_mutation_rate=0.1,
    final_mutation_rate=0.01,
    total_generations=100
)

population = ga.initial_population
average_fitnesses = []
best_fitnesses = []

for generation in (pbar := tqdm(range(ga.total_generations))):

    population = ga.evolve(
        population=population,
        fitnesses=evaluate_fitness(
            population=population, ps_params=plant_params, prices=example
        ),
    )
    average_fitnesses.append(ga.average_fitness)
    best_fitnesses.append(ga.best_fitness)

    pbar.set_description(f"Best fitness: {best_fitnesses[-1]}")

ga.best_fitness

275136.0

ga.best_dna

array([ 0., -1., -1., -1., -1., -1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
        0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0., -1.,
       -1., -1., -1., -1., -1.,  0.,  0.,  0.,  0.,  0.,  0., -1.,  0.,
        0.,  0.,  1.,  0.,  0.,  0.,  0.,  0., -1., -1., -1., -1., -1.,
        0.,  0.,  0.,  1.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
        0.,  0.,  0.,  0.,  0.,  0.,  0., -1., -1.,  0.,  0.,  0.,  0.,
        0.,  1.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,
        0.,  0.,  0.,  0., -1.,  0., -1., -1., -1., -1.,  0.,  0.,  0.,
        0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,
        0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,
        1.,  0.,  0.,  0.,  0.,  1.,  0.,  1.,  1.,  0.,  0.,  0.,  0.,
        0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  1.,
        0.,  0.,  0.,  0.,  1.,  1.,  0.,  0.,  0.,  0.,  0.,  0.])

(
    pn.ggplot(
        data=(
            pd.DataFrame(
                {
                    "average": average_fitnesses,
                    "best": best_fitnesses,
                    "generation": np.arange(1, len(average_fitnesses) + 1),
                }
            )
            # .tail(len(average_fitnesses)-10)
            .melt(id_vars="generation")
        ),
        mapping=pn.aes(x="generation", y="value", colour="variable"),
    )
    + pn.geom_line()
    + pn.theme(figure_size=[6, 3])
    # + pn.scale_y_log10()
)

best_profile = (
    example.to_pandas().assign(
        action=ga.best_dna,
        colour_id=lambda x: np.select(
            [x.action == 1, x.action == 0, x.action == -1],
            ["pump", "nothing", "turbine"],
        ),
        water_level=lambda x: plant_params["INITIAL_WATER_LEVEL_PCT"] * plant_params["MAX_STORAGE_M3"]
        + np.select(
            [x.action == 1, x.action == -1],
            [plant_params["PUMP_RATE_M3H"], -plant_params["TURBINE_RATE_M3H"]],
            0,
        ).cumsum(),
    )
)

best_profile.head()

	spot	utc_time	action	colour_id	water_level
0	101.540	2022-01-01 00:00:00+00:00	0.000	nothing	0.000
1	52.130	2022-01-01 01:00:00+00:00	-1.000	turbine	-500.000
2	20.780	2022-01-01 02:00:00+00:00	-1.000	turbine	-1000.000
3	15.660	2022-01-01 03:00:00+00:00	-1.000	turbine	-1500.000
4	21.470	2022-01-01 04:00:00+00:00	-1.000	turbine	-2000.000

Tuning Hyperparameters

def genetic_algorithm_tuning(trial):

    config = {
        "dna_size": window_size,
        "discrete_action_space": [-1, 0, 1],
        "elitism": trial.suggest_float("elitism", 0.1, 0.9),
        "population_size": trial.suggest_categorical("population_size", [50, 100, 500, 1000]),
        "survival_rate": trial.suggest_float("survival_rate", 0.1, 0.9),
        "initial_mutation_rate": trial.suggest_float("initial_mutation_rate", 0.2, 1),
        "final_mutation_rate": 0.01,
        "total_generations": 100,
    }

    ga = GA_discrete_actions(**config)

    population = ga.initial_population

    for _ in range(config["total_generations"]):
        population = ga.evolve(
            population=population,
            fitnesses=evaluate_fitness(
                population=population, ps_params=plant_params, prices=example
            ),
        )

    return ga.best_fitness

np.random.seed(42)

study = optuna.create_study(direction='maximize')
study.optimize(genetic_algorithm_tuning, n_trials=1e4, timeout=60*60)

study.best_params

with open("02 - Tuning Results/ga_tuning_params.json", "w") as outfile: 
    json.dump(study.best_params, outfile)

Final Run

So, let’s do a final run with the best hyperparameters found from tuning for a longer time period and see where we end.

with open("02 - Tuning Results/ga_tuning_params.json") as infile: 
    ga_params = json.load(infile)

np.random.seed(42)
ga = GA_discrete_actions(
    dna_size=window_size,
    discrete_action_space=[-1, 0, 1],
    elitism=ga_params["elitism"],
    population_size=ga_params["population_size"],
    survival_rate=ga_params["survival_rate"],
    initial_mutation_rate=ga_params["initial_mutation_rate"],
    final_mutation_rate=0.1,
    total_generations=250,
)

population = ga.initial_population
params_best = None
fitness_best = None
average_fitnesses = []
best_fitnesses = []
best_solutions = []
all_fitnesses = []

for generation in (pbar := tqdm(range(ga.total_generations))):
    fitnesses = evaluate_fitness(
        population=population, ps_params=plant_params, prices=example
    )
    population = ga.evolve(
        population=population,
        fitnesses=fitnesses,
    )

    # Store historically best solution
    if params_best is None or ga.best_fitness > fitness_best:
        params_best = ga.best_dna
        fitness_best = ga.best_fitness

    # Store all solutions for animation
    best_solutions.append(ga.best_dna)
    all_fitnesses.append(fitnesses)

    # Store history for visualisation
    average_fitnesses.append(ga.average_fitness)
    best_fitnesses.append(fitness_best)

    pbar.set_description(f"Best fitness: {best_fitnesses[-1]:.0f}")

ga.best_fitness

339890.0

fitness_best

339890.0

Store the best solution found for later comparison and visualisation:

actions = example.with_columns(ga_action = pl.Series(params_best))

actions.head()

actions.to_pandas().to_csv("./03 - Final Run Results/actions.csv", index=False)

with open('03 - Final Run Results/ga_actions.npy', 'wb') as f:
    np.save(f, np.array(best_solutions))

with open('03 - Final Run Results/ga_best_fitnesses.npy', 'wb') as f:
    np.save(f, np.array(best_fitnesses))

with open('03 - Final Run Results/ga_all_fitnesses.npy', 'wb') as f:
    np.save(f, np.array(all_fitnesses))

Simulated Annealing

The Energy Function

def profit(actions, ps_params, prices):
    water_level = ps_params["INITIAL_WATER_LEVEL"]
    profit = 0

    for action, price in zip(actions, prices["spot"]):
        # Pump (-1)
        if action == -1:
            if (
                (water_level + ps_params["PUMP_RATE_M3H"])
                <= ps_params["MAX_STORAGE_M3"]
            ):
                profit -= ps_params["PUMP_POWER_MW"] * price
                water_level += ps_params["PUMP_RATE_M3H"]
            else:
                profit -= 1e6
        # Turbine (1)
        if action == 1:
            if (
                (water_level - ps_params["TURBINE_RATE_M3H"])
                >= ps_params["MIN_STORAGE_M3"]
            ):
                profit += ps_params["TURBINE_POWER_MW"] * price
                water_level -= ps_params["TURBINE_RATE_M3H"]
            else:
                profit -= 1e6
    
    return profit

change current solution slightly (neighbour state)
accept new state with probability that depends on the energy of both states and on temperature parameters
- smaller energy is better than larger energy
- probability function must be positive even if the new state has higher energy (is worse)
  - this helps escape local optima

(
    pd.DataFrame({"iter": np.arange(0, 100)}).assign(
        temp=lambda x: 1 - (x.iter + 1) / 100,
        acceptance_prob=lambda x: np.exp(
            - (0.5) / (x.temp + 0.0001)
        ),
    )
    .plot(x="temp", y="acceptance_prob", ylim=[0,1])
)

flip_size = 1
iterations = 100
initial_temp = 10000
temp_decay = 0.5
params = np.random.choice([-1,0,1], size=example.shape[0])
params_best = None
energy_best = None

for i in (pbar := tqdm(range(iterations))):
    
    # Modify temperature parameters
    # temperature = initial_temp/(i + 1)
    temperature = initial_temp - i/iterations * initial_temp

    # Generate random neighbour by randomly flipping actions
    params_new = params.copy()
    for _ in range(flip_size):
        idx = np.random.choice(np.arange(0, len(params)))
        new_options = [x for x in [-1,0,1] if x != params[idx]]
        params_new[idx] = np.random.choice(new_options)
        
    # Calculate energy for old and new state
    energy_old = -profit(actions=params, prices=prices, ps_params=plant_params)
    energy_new = -profit(actions=params_new, prices=prices, ps_params=plant_params)

    # Store best solution
    if params_best is None or energy_new < energy_best:
        params_best = params_new
        energy_best = energy_new

    # Calculate acceptance probability
    if energy_new <= energy_old:
        params = params_new
    else:
        acceptance_prob = np.exp(-(energy_new - energy_old)/(temperature + 1e-8))
        
        if np.random.uniform() < acceptance_prob:
            params = params_new

    pbar.set_description(f"Profit: {-energy_old}")

Tuning Hyperparameters

def simulated_annealing(config, tune_mode=False):
    
    params = np.random.choice([-1,0,1], size=example.shape[0])
    params_best = None
    energy_best = None

    for i in (pbar := tqdm(range(config["iterations"]))):
        
        # Linear temperature decay (easier to scale to larger number of iterations later)
        temperature = config["initial_temp"] - i/config["iterations"] * config["initial_temp"]

        # Generate random neighbour by randomly flipping actions
        params_new = params.copy()
        for _ in range(config["flip_size"]):
            idx = np.random.choice(np.arange(0, len(params)))
            new_options = [x for x in [-1,0,1] if x != params[idx]]
            params_new[idx] = np.random.choice(new_options)
            
        # Calculate energy for old and new state
        energy_old = -profit(actions=params, prices=example, ps_params=plant_params)
        energy_new = -profit(actions=params_new, prices=example, ps_params=plant_params)
        if energy_best is None:
            params_best = params
            energy_best = energy_old
                
        # Calculate acceptance probability
        # If solution is better than previous accept immediately
        if energy_new < energy_old:
            # Solution is better than previous, but is it better than best
            if energy_new < energy_best:
                params_best = params_new.copy()
                energy_best = energy_new
            params = params_new.copy()
        # If solution isn't better than previous, accept maybe
        else:
            acceptance_prob = np.exp(-(energy_new - energy_old)/(temperature + 1e-8))
            if np.random.uniform() < acceptance_prob:
                params = params_new.copy()

        if tune_mode:
            train.report({"energy": energy_best})
        else:
            pbar.set_description(f"Profit: {-energy_best:.2f}")

    if not tune_mode:
        return params_best

# Need this line for locally defined modules to work with ray
# ray.init(runtime_env={"working_dir": "."}, ignore_reinit_error=True)
np.random.seed(42)

analysis = tune.run(
    tune.with_parameters(
        simulated_annealing,
        tune_mode=True,
    ),
    config={
        "iterations": 1000,
        "initial_temp": tune.uniform(0.01, 10),
        "flip_size": tune.randint(1, 50),
    },
    metric="energy",
    mode="min",
    local_dir="tune_results",
    name="SA",
    search_alg=OptunaSearch(),
    time_budget_s=60*60,
    num_samples=-1,
    trial_dirname_creator=lambda trial: f"{trial.trainable_name}_{trial.trial_id}",
)

f"Best Profit: {-analysis.best_result['energy']:.2f}"

top_runs = analysis.dataframe().sort_values("energy", ascending=True).head(10)["trial_id"].to_list()

fig = px.line(
    data_frame=pd.concat(analysis.trial_dataframes.values()).query(
        "trial_id in @top_runs"
    ),
    x="training_iteration",
    y="energy",
    color="trial_id",
)
fig.show()

analysis.best_config

with open("02 - Tuning Results/sa_tuning_params.json", "w") as outfile: 
    json.dump(analysis.best_config, outfile)

Final Run

Here we need to adjust the hyperparameter of temperature decay

TBD: Explain why the formula doesn’t work (float precision) in practice

init_new = init_old * d^x_old / d^x_new

with open("02 - Tuning Results/sa_tuning_params.json") as infile: 
    sa_params = json.load(infile)
old_sa_params = sa_params.copy()
old_sa_params

{'iterations': 1000, 'initial_temp': 7.629941060448957, 'flip_size': 2}

sa_params["iterations"] = 75_000
sa_params["initial_temp"] *= sa_params["iterations"] / old_sa_params["iterations"]

sa_params

{'iterations': 75000, 'initial_temp': 572.2455795336717, 'flip_size': 2}

np.random.seed(42)
params = np.random.choice([-1,0,1], size=example.shape[0])
params_best = None
energy_best = None
best_profits = []
best_solutions = []
all_profits = []


for i in (pbar := tqdm(range(sa_params["iterations"]))):
    
    # Modify temperature parameters
    # temperature = initial_temp/(i + 1)
    temperature = sa_params["initial_temp"] - i/sa_params["iterations"] * sa_params["initial_temp"]

    # Generate random neighbour by randomly flipping actions
    params_new = params.copy()
    for _ in range(sa_params["flip_size"]):
        idx = np.random.choice(np.arange(0, len(params)))
        new_options = [x for x in [-1,0,1] if x != params[idx]]
        params_new[idx] = np.random.choice(new_options)
        
    # Calculate energy for old and new state
    energy_old = -profit(actions=params, prices=example, ps_params=plant_params)
    all_profits.append(-energy_old)
    energy_new = -profit(actions=params_new, prices=example, ps_params=plant_params)
    if energy_best is None:
        params_best = params
        energy_best = energy_old
            
    # Calculate acceptance probability
    # If solution is better than previous accept immediately
    if energy_new < energy_old:
        # Solution is better than previous, but is it better than best
        if energy_new < energy_best:
            params_best = params_new.copy()
            energy_best = energy_new
        params = params_new.copy()
    # If solution isn't better than previous, accept maybe
    else:
        acceptance_prob = np.exp(-(energy_new - energy_old)/(temperature + 1e-8))
        if np.random.uniform() < acceptance_prob:
            params = params_new.copy()

    # Save interim results
    best_solutions.append(params_best)
    best_profits.append(-energy_best)
    

    pbar.set_description(f"Profit: {-energy_best:,.0f} Temperature: {temperature:,.2f}")

profit(actions=params, prices=example, ps_params=plant_params)

317164.0

actions = (
    pl.read_csv("./03 - Final Run Results/actions.csv", try_parse_dates=True)
    .select("utc_time", "spot", "ga_action")
    .with_columns(pl.col("ga_action").cast(pl.Int64))
    .with_columns(sa_action = pl.Series(params).cast(pl.Int64))
)
actions.to_pandas().to_csv("./03 - Final Run Results/actions.csv", index=False)
actions.head()

with open('03 - Final Run Results/sa_actions.npy', 'wb') as f:
    np.save(f, np.array(best_solutions))

with open('03 - Final Run Results/sa_best_profits.npy', 'wb') as f:
    np.save(f, np.array(best_profits))

with open('03 - Final Run Results/sa_all_profits.npy', 'wb') as f:
    np.save(f, np.array(all_profits))

Brute Force Search

np.random.seed(42)
params_best = None
profit_best = None
best_profits = []
best_solutions = []


for i in (pbar := tqdm(range(150_000))):

    # Generate random candidate
    params = np.random.choice(a=[-1, 0, 1], size=example.shape[0])

    # Calculate profit
    current_profit = profit(actions=params, prices=example, ps_params=plant_params)

    # Check if it's historically best
    if profit_best is None or current_profit > profit_best:
        params_best = params.copy()
        profit_best = current_profit

    # Save interim results
    best_solutions.append(params_best)
    best_profits.append(profit_best)

    pbar.set_description(f"Profit: {profit_best:.2f}")

actions = ( 
    pl.read_csv("./03 - Final Run Results/actions.csv", try_parse_dates=True)
    .select("utc_time", "spot", "ga_action", "sa_action")
    .with_columns(pl.col("ga_action").cast(pl.Int64))
    .with_columns(pl.col("sa_action").cast(pl.Int64))
    .with_columns(bf_action = pl.Series(params_best).cast(pl.Int64))
)
actions.to_pandas().to_csv("./03 - Final Run Results/actions.csv", index=False)
actions.head()

with open('03 - Final Run Results/bf_actions.npy', 'wb') as f:
    np.save(f, np.array(best_solutions))

with open('03 - Final Run Results/bf_best_profits.npy', 'wb') as f:
    np.save(f, np.array(best_profits))

Mixed Integer Linear Programming

# Create names for decision variables
dec_vars = range(example.shape[0])

# The cost/revenue associated with each decision variable is the
# spot price
spot = dict(zip(dec_vars, example["spot"].to_list()))

# Decision variables
dp = LpVariable.dicts("pump_dummy", dec_vars, 0, 1, cat="Binary")
dt = LpVariable.dicts("turbine_dummy", dec_vars, 0, 1, cat="Binary")

# State Variables
water_level = LpVariable.dict(
    "water_level",
    dec_vars,
    plant_params["MIN_STORAGE_M3"],
    plant_params["MAX_STORAGE_M3"],
    cat="Continuous",
)

# Initialise Model
model = LpProblem(name="pumped_storage_optimisation", sense=LpMaximize)

# Add objective function
model += (
    lpSum(
        [
            dt[i] * plant_params["TURBINE_POWER_MW"] * example["spot"][i]
            - dp[i] * plant_params["PUMP_POWER_MW"] * example["spot"][i]
            for i in dec_vars
        ]
    ),
    "revenue",
)

# Add constraints
for i in dec_vars:
    # Don't pump and turbine at the same time
    model += (dt[i] + dp[i] <= 1, f"no_simultaneous_pump_and_turbine_upper_{i}")
    model += (dt[i] + dp[i] >= 0, f"no_simultaneous_pump_and_turbine_lower_{i}")

    # Add constraint to update water level
    if i == 0:
        model += (
            water_level[i]
            == plant_params["INITIAL_WATER_LEVEL"]
            + dp[i] * plant_params["PUMP_RATE_M3H"]
            - dt[i] * plant_params["TURBINE_RATE_M3H"],
            f"initial_water_level_{i}",
        )
    else:
        # Update water level based on previous level, pumping, and turbine action
        model += (
            water_level[i]
            == water_level[i - 1]
            + dp[i] * plant_params["PUMP_RATE_M3H"]
            - dt[i] * plant_params["TURBINE_RATE_M3H"],
            f"water_level_update_{i}",
        )

    # Add constraints for water level boundaries
    model += (water_level[i] <= plant_params["MAX_STORAGE_M3"], f"max_storage_{i}")
    model += (water_level[i] >= plant_params["MIN_STORAGE_M3"], f"min_storage_{i}")


# Solve the model
model.solve()

LpStatus[model.status]

'Optimal'

model.objective.value()

350291.0

best_profile = (
    pd.DataFrame(
        {
            "variable": [v.name for v in model.variables()],
            "value": [v.value() for v in model.variables()],
        }
    )
    .assign(
        name=lambda x: x["variable"].apply(lambda r: r.rsplit("_", 1)[0]),
        hour=lambda x: x["variable"].apply(lambda r: r.rsplit("_", 1)[1]).astype(int),
    )
    .drop("variable", axis=1)
    .pivot(index="hour", columns="name", values="value")
    .reset_index()
    .rename_axis(None, axis=1)
    .sort_values("hour")
    .assign(
        action=lambda x: np.select(
            condlist=[
                x["pump_dummy"] == 1,
                x["turbine_dummy"] == 1,
            ],
            choicelist=[
                -1,
                1,
            ],
            default=0,
        ),
        colour_id=lambda x: np.select(
            condlist=[
                x["pump_dummy"] == 1,
                x["turbine_dummy"] == 1,
            ],
            choicelist=[
                "pump",
                "turbine",
            ],
            default="nothing",
        ),
        utc_time=pd.to_datetime(example["utc_time"]),
        spot=example["spot"].to_list(),
    )
    .drop(["pump_dummy", "turbine_dummy", "hour"], axis=1)
)

best_profile.describe()

actions = (
    pl.read_csv("./03 - Final Run Results/actions.csv", try_parse_dates=True)
    .select("utc_time", "spot", "ga_action", "sa_action", "bf_action")
    .with_columns(pl.col("ga_action").cast(pl.Int64))
    .with_columns(pl.col("sa_action").cast(pl.Int64))
    .with_columns(pl.col("bf_action").cast(pl.Int64))
    .with_columns(milp_action = pl.Series(best_profile["action"]).cast(pl.Int64))
)
actions.to_pandas().to_csv("./03 - Final Run Results/actions.csv", index=False)
actions.head()

Comparison of Best Solutions

actions = (
    pl.read_csv("./03 - Final Run Results/actions.csv", try_parse_dates=True)
    .with_columns(
        pl.col("ga_action").cast(pl.Int64),
        pl.col("sa_action").cast(pl.Int64),
        pl.col("bf_action").cast(pl.Int64),
        pl.col("milp_action").cast(pl.Int64),
    )
)
actions.head()

shape: (5, 6)

utc_time	spot	ga_action	sa_action	bf_action	milp_action
datetime[μs, UTC]	f64	i64	i64	i64	i64
2022-01-01 00:00:00 UTC	101.54	0	0	-1	0
2022-01-01 01:00:00 UTC	52.13	-1	-1	0	-1
2022-01-01 02:00:00 UTC	20.78	-1	-1	-1	-1
2022-01-01 03:00:00 UTC	15.66	-1	-1	0	-1
2022-01-01 04:00:00 UTC	21.47	-1	-1	1	-1

algos = ["ga_action", "sa_action", "bf_action", "milp_action"]
algo_names = ["Genetic Algorithm", "Simulated Annealing", "Random Search", "Mixed Integer Linear Programming"]

for idx, algo in enumerate(algos):

    best_profile = (
        actions.select("utc_time", "spot", algo)
        .rename({algo: "action"})
        .with_columns(
            colour_id=pl.when(pl.col("action") == -1)
            .then(pl.lit("pump"))
            .when(pl.col("action") == 1)
            .then(pl.lit("turbine"))
            .otherwise(pl.lit("Nothing"))
        )
        .with_columns(
            water_level=plant_params["INITIAL_WATER_LEVEL_PCT"]
            + (
                pl.when(pl.col("action") == -1)
                .then(plant_params["PUMP_RATE_M3H"])
                .when(pl.col("action") == 1)
                .then(-plant_params["TURBINE_RATE_M3H"])
                .otherwise(0)
                .cum_sum()
            )
        )
        .to_pandas()
        .sort_values("utc_time")
    )

    best_profit = profit(actions=best_profile["action"], prices=example, ps_params=plant_params)

    # Operating Profile
    fig = (
        pn.ggplot(
            data=best_profile.assign(
                spot=lambda x: (x.spot - x.spot.mean()) / x.spot.std()
            ),
            mapping=pn.aes(x="utc_time"),
        )
        + pn.geom_line(pn.aes(y="spot"))
        + pn.geom_point(pn.aes(y="spot"))
        + pn.geom_col(pn.aes(y="action", fill="colour_id"), alpha=0.5)
        + pn.labs(title=f"Best found operating profile for {algo_names[idx]}: {best_profit:,.0f} CHF")
        + pn.scale_fill_manual(values=["black", "firebrick", "midnightblue"])
        + pn.geom_hline(yintercept=0, linetype="dotted")
        # + pn.scale_x_date(date_breaks="1 month", date_labels="%d/%m")
        + pn.theme(figure_size=[10, 3], legend_title=pn.element_blank())
    )
    display(fig)

    # Lake level
    fig = (
        pn.ggplot(data=best_profile, mapping=pn.aes(x="utc_time", y="water_level"))
            + pn.geom_line()
            + pn.labs(title=f"Lake Levels for {algo_names[idx]}")
            + pn.geom_hline(yintercept=plant_params["MAX_STORAGE_M3"], linetype="dotted")
            + pn.geom_hline(yintercept=plant_params["MIN_STORAGE_M3"], linetype="dotted")
            # + pn.scale_x_date(date_breaks="1 month", date_labels="%d/%m")
            # + pn.coord_cartesian(ylim=[0, best_profile.water_level.max()*1.1])
            + pn.theme(figure_size=[10, 3], legend_title=pn.element_blank())
        )
    display(fig)

Conclusion

TBD

Animation

If you made it this far, thank you very much for reading! This section of the blog is for the interested who want to know how I made the animation at the start of the blog.

I wanted to show:

The current state of the optimisation at every stage for SA and GA
The progress over the allowed time slot (30 seconds)

Static Plot Generation

GA All Fitnesses

with open("./03 - Final Run Results/ga_all_fitnesses.npy", "rb") as f:
    ga_all_fitnesses = np.load(f)

np.random.seed(1)
ga_all_fitnesses_df = (
    pl.DataFrame(
        ga_all_fitnesses, schema=[f"{i}" for i in range(ga_all_fitnesses.shape[0])]
    )
    .unpivot(variable_name="generation")
    .with_columns(pl.col("generation").cast(pl.Int64))
    .sample(fraction=1)
)

# warnings.filterwarnings('ignore')
plot_data = None

for idx in tqdm(range(ga_all_fitnesses.shape[0])):

    # Filter data
    plot_data = (
        pl.DataFrame(
            ga_all_fitnesses, schema=[f"{i}" for i in range(ga_all_fitnesses.shape[0])]
        )
        .unpivot(variable_name="generation")
        .with_columns(pl.col("generation").cast(pl.Int64))
        .filter(
            (pl.col("generation") <= idx) 
            # & (pl.col("generation") >= idx - 50)
        )
        .with_columns(
            # shading=((pl.col("generation") + 1) / (idx + 1)) ** 1e600
            shading=pl.when(pl.col("generation") == idx).then(pl.lit(1)).otherwise(pl.lit(0.01))
        )

    )
    # print(plot_data)

    # Plot data
    fig = (
        pn.ggplot(
            data=plot_data.group_by("generation").head(50),
            mapping=pn.aes(x="generation", y="value"),
        )
        + pn.geom_point(pn.aes(alpha="shading"))
        + pn.labs(title=f"Genetic Algorithm: Population At Generation {idx}",
                  y = "profit", x = "generation")
        + pn.scale_x_continuous(limits=(0, ga_all_fitnesses.shape[0]))
        + pn.scale_y_continuous(
            labels=comma_format(), limits=[-4e5, 4e5]
        )
        # + pn.theme_light()
        + pn.theme(
            figure_size=[7, 7 * ((5/3) / (4/2))], legend_title=pn.element_blank(), 
            legend_position="none"
        )
    )
    fig.save(filename = f"./04 - Animation/GA_fitnesses/generation_{idx:03d}", height=7 * ((5/3) / (4/2)), width=7, dpi=300)
    # display(fig)
    # clear_output(wait=True)
    # break

SA All Fitnesses

with open("./03 - Final Run Results/sa_all_profits.npy", "rb") as f:
    sa_all_profits = np.load(f)

Reduce to 1000 frames to keep the folder manageable:

max_frames = 1000
plot_data = (
    pl.DataFrame(pl.Series(values=sa_all_profits, name="profit")).with_columns(pl.Series(np.arange(0, len(sa_all_profits))).alias("iteration"))
    .with_columns(((pl.col("iteration") / len(sa_all_profits)) * max_frames).floor().alias("frame"))
    .group_by("frame")
    .tail(1)
    .sort("frame")
)
print(plot_data.shape)
plot_data.head()

(1000, 3)

shape: (5, 3)

frame	profit	iteration
f64	f64	i32
0.0	-1.195824e6	74
1.0	-1.106363e6	149
2.0	-63354.0	224
3.0	9623.0	299
4.0	35701.0	374

for idx in tqdm(range(max_frames)):

    # Plot data
    fig = (
        pn.ggplot(
            data=plot_data.filter(pl.col("frame") <= idx),
            mapping=pn.aes(x="iteration", y="profit"),
        )
        + pn.geom_line()
        + pn.labs(
            title=f"Simulated Annealing: Solution At Iteration {plot_data.filter(pl.col('frame') == idx)['iteration'][0]}",
            y="profit",
            x="iteration",
        )
        + pn.scale_x_continuous(
            labels=comma_format(), limits=(0, plot_data["iteration"].max())
        )
        + pn.scale_y_continuous(labels=comma_format())
        + pn.coord_cartesian(
            ylim=[
                plot_data.filter(pl.col("frame") <= idx).tail(500)["profit"].min(),
                plot_data.filter(pl.col("frame") <= idx).tail(500)["profit"].max(),
            ]
        )
        + pn.theme(
            figure_size=[7, 7 * ((5/3) / (4/2))], legend_title=pn.element_blank(), legend_position="none"
        )
    )
    fig.save(filename = f"./04 - Animation/SA_fitnesses/iteration_{idx:04d}", height=7 * ((5/3) / (4/2)), width=7, dpi=300)
    # display(fig)
    # clear_output(wait=True)
    # break

GA Profiles

with open("./03 - Final Run Results/ga_actions.npy", "rb") as f:
    ga_actions = np.load(f)

for idx in tqdm(range(ga_actions.shape[0])):

    plot_data = (
        example.select("utc_time", "spot")
        .with_columns(action = pl.Series(ga_actions[idx]))
        .with_columns(
            colour_id=pl.when(pl.col("action") == -1)
            .then(pl.lit("pump"))
            .when(pl.col("action") == 1)
            .then(pl.lit("turbine"))
            .otherwise(pl.lit("Nothing"))
        )
        .with_columns(
            water_level=plant_params["INITIAL_WATER_LEVEL_PCT"]
            + (
                pl.when(pl.col("action") == -1)
                .then(plant_params["PUMP_RATE_M3H"])
                .when(pl.col("action") == 1)
                .then(-plant_params["TURBINE_RATE_M3H"])
                .otherwise(0)
                .cum_sum()
            )
        )
        .to_pandas()
        .sort_values("utc_time")
    )
    
    # Operating Profile
    fig = (
        pn.ggplot(
            data=plot_data.assign(
                spot=lambda x: (x.spot - x.spot.mean()) / x.spot.std()
            ),
            mapping=pn.aes(x="utc_time"),
        )
        + pn.geom_line(pn.aes(y="spot"))
        + pn.geom_point(pn.aes(y="spot"))
        + pn.geom_col(pn.aes(y="action", fill="colour_id"), alpha=0.5)
        + pn.labs(title=f"Genetic Algorithm: Best Profile At Generation {idx}", x = " ")
        + pn.scale_fill_manual(values=["black", "firebrick", "midnightblue"])
        + pn.geom_hline(yintercept=0, linetype="dotted")
        # + pn.scale_x_date(date_breaks="1 month", date_labels="%d/%m")
        + pn.theme(
            figure_size=[7, 7 * ((5/3) / (4/2))],
            legend_title=pn.element_blank(),
            legend_position="none",
            axis_text_y=pn.element_blank(),
        )
    )
    fig.save(filename = f"./04 - Animation/GA_profiles/generation_{idx:04d}", height=7 * ((5/3) / (4/2)), width=7, dpi=300)
    # display(fig)
    # clear_output(wait=True)
    # break

SA Profiles

with open("./03 - Final Run Results/sa_actions.npy", "rb") as f:
    sa_actions = np.load(f)

max_frames = 1000
frames = [i for i in range(len(sa_actions)) if i % int(len(sa_actions)/max_frames) == 0]

for idx in tqdm(range(max_frames)):

    plot_data = (
        example.select("utc_time", "spot")
        .with_columns(action=pl.Series(sa_actions[frames[idx]]))
        .with_columns(
            colour_id=pl.when(pl.col("action") == -1)
            .then(pl.lit("pump"))
            .when(pl.col("action") == 1)
            .then(pl.lit("turbine"))
            .otherwise(pl.lit("Nothing"))
        )
        .with_columns(
            water_level=plant_params["INITIAL_WATER_LEVEL_PCT"]
            + (
                pl.when(pl.col("action") == -1)
                .then(plant_params["PUMP_RATE_M3H"])
                .when(pl.col("action") == 1)
                .then(-plant_params["TURBINE_RATE_M3H"])
                .otherwise(0)
                .cum_sum()
            )
        )
        .to_pandas()
        .sort_values("utc_time")
    )

    # Operating Profile
    fig = (
        pn.ggplot(
            data=plot_data.assign(
                spot=lambda x: (x.spot - x.spot.mean()) / x.spot.std()
            ),
            mapping=pn.aes(x="utc_time"),
        )
        + pn.geom_line(pn.aes(y="spot"))
        + pn.geom_point(pn.aes(y="spot"))
        + pn.geom_col(pn.aes(y="action", fill="colour_id"), alpha=0.5)
        + pn.labs(
            title=f"Simulated Annealing: Best Profile At Iteration {frames[idx]}", x=" "
        )
        + pn.scale_fill_manual(values=["black", "firebrick", "midnightblue"])
        + pn.geom_hline(yintercept=0, linetype="dotted")
        # + pn.scale_x_date(date_breaks="1 month", date_labels="%d/%m")
        + pn.theme(
            figure_size=[7, 7 * ((5/3) / (4/2))],
            legend_title=pn.element_blank(),
            legend_position="none",
            axis_text_y=pn.element_blank(),
        )
    )
    fig.save(filename = f"./04 - Animation/SA_profiles/generation_{idx:04d}", height=7 * ((5/3) / (4/2)), width=7, dpi=300)
    # display(fig)
    # clear_output(wait=True)
    # break

Best Profits

with open("./03 - Final Run Results/ga_best_fitnesses.npy", "rb") as f:
    ga_profits = np.load(f)

with open("./03 - Final Run Results/sa_best_profits.npy", "rb") as f:
    sa_profits = np.load(f)

with open("./03 - Final Run Results/bf_best_profits.npy", "rb") as f:
    bf_profits = np.load(f)

len(ga_profits)

len(sa_profits)

len(bf_profits)

max_frames = len(ga_profits)
sa_frames = [i-1 for i in range(1, len(sa_profits) + 1) if i % int(len(sa_profits)/max_frames) == 0]
bf_frames = [i-1 for i in range(1, len(bf_profits) + 1) if i % int(len(bf_profits)/max_frames) == 0]

plot_data = pl.DataFrame(
    {
        "generation": np.arange(0, len(ga_profits)),
        "Genetic Algorithm": ga_profits,
        "Simulated Annealing": sa_profits[sa_frames],
        "Brute Force Random Search": bf_profits[bf_frames],
    }
).unpivot(index="generation")
plot_data.head()

shape: (5, 3)

generation	variable	value
i32	str	f64
0	"Genetic Algorithm"	-177561.0
1	"Genetic Algorithm"	-19350.0
2	"Genetic Algorithm"	-12588.0
3	"Genetic Algorithm"	-12588.0
4	"Genetic Algorithm"	23985.0


#| eval: false
for idx in tqdm(range(plot_data["generation"].unique().shape[0])):
    # Operating Profile
    fig = (
        pn.ggplot(
            data=plot_data.filter(pl.col("generation") <= idx),
            mapping=pn.aes(x="generation", y="value", colour="variable"),
        )
        + pn.geom_line()
        + pn.labs(y = "Best Found Profit", x="Time")
        + pn.scale_x_continuous(limits=[0, plot_data["generation"].max()])
        + pn.scale_y_continuous(labels=comma_format())
        + pn.coord_cartesian(
            ylim=[
                plot_data.filter(pl.col("generation") <= idx).tail(20)["value"].min(),
                plot_data["value"].max(),
            ]
        )
        # + pn.scale_fill_manual(values=["black", "firebrick", "midnightblue"])
        + pn.theme(
            figure_size=[10, 10 * ((5/3) / 4)],
            legend_title=pn.element_blank(),
            axis_text_x=pn.element_blank(),
            # legend_position="none",
        )
    )
    fig.save(filename = f"./04 - Animation/Profits/generation_{idx:03d}", height=10 * ((5/3) / 4), width=10, dpi=300)
    # display(fig)
    # clear_output(wait=True)
    # break

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Video Generation

I will create videos from the raw plots and later edit them together in a video editing program manually for the final animation.

def generate_video(image_path, video_path, fps):
    
    # Get list of PNG files sorted
    file_names = sorted([f for f in os.listdir(image_path) if f.endswith('.png')])

    writer = imageio.get_writer(video_path, fps=fps)
    # Read PNG files and append to images list
    for file_name in file_names:
        file_path = os.path.join(image_path, file_name)
        im = imageio.v2.imread(file_path)
        writer.append_data(im)

    writer.close()

GA All profits

generate_video(
    image_path="./04 - Animation/GA_fitnesses/",
    video_path="./04 - Animation/ga_fitnesses.mp4",
    fps=15
)

SA All Profits

generate_video(
    image_path="./04 - Animation/SA_fitnesses/",
    video_path="./04 - Animation/sa_fitnesses.mp4",
    fps=60
)

GA Profiles

generate_video(
    image_path="./04 - Animation/GA_profiles/",
    video_path="./04 - Animation/ga_profiles.mp4",
    fps=15
)

SA Profiles

generate_video(
    image_path="./04 - Animation/SA_profiles/",
    video_path="./04 - Animation/sa_profiles.mp4",
    fps=60
)

Best Profits

generate_video(
    image_path="./04 - Animation/Profits/",
    video_path="./04 - Animation/best_profits.mp4",
    fps=15
)